Worksheet Solutions: Patterns | Mathematics for Class 5 PDF Download

Question 1: Find the next term in the given patterns:

• 204, 305, 406 ..........
• ABC, CDE, EF , ..........
• Aa, Bb, Cc, Dd, Ee, ..........
• 1, 2, 3, 4W, ..........
• A, A, A, AW, ..........

Ans:

• 204, 305, 406, ...
  Pattern Analysis: Each number seems to increment the hundreds digit by 1 and the tens digit by 0, while the units digit increases by 1.
  Next Term: Following this pattern, after 406 (4, 0, 6), the next number should be 507 (5, 0, 7).
• ABC, CDE, EF, ...
  Pattern Analysis: Each group shifts up by 2 letters in the alphabet. ABC (A, B, C); CDE (C, D, E); EF (E, F).
  Next Term: After EF, if we continue shifting by 2, the next term would start at G. Considering only the next two letters, the sequence would be GH.
• Aa, Bb, Cc, Dd, Ee, ...
  Pattern Analysis: This pattern simply progresses through the alphabet with each letter followed by its lowercase version.
  Next Term: The sequence naturally continues to the next letter, so after Ee comes Ff.
• 1, 2, 3, 4W, ...
  Pattern Analysis: This sequence consists of consecutive numbers until 4, after which a 'W' is added.
  Next Term: It seems logical to continue the pattern of adding consecutive numbers and then appending 'W' after 4. Thus, the next term would be 5W.
• A, A, A, AW, ...
  Pattern Analysis: The letter A is repeated three times, with a 'W' appended to the fourth term.
  Next Term: Assuming the pattern of appending 'W' continues every fourth term, the next term would be A.

Question 2: Fill in the missing numbers in each to get a magic triangle. Also, find the magic number:

Ans: To solve the Magic Triangle using the numbers 8 to 13, we first calculate the total sum of these numbers and then determine the magic number needed for each side of the triangle.
Step 1: Calculate Total Sum and Average Magic Number
The numbers to be used are 8, 9, 10, 11, 12, and 13. The sum of these numbers is calculated as follows:
Sum = 8 + 9 + 10 + 11 + 12 + 13 = 63
Magic Number = Sum / 3 sides = 63 / 3 = 21
Step 2: Determine Missing Numbers
With the given numbers 9, 13, and 8:
One side already correctly sums to 21 (13 + 8).
The other sides must also sum to 21, needing the numbers 10, 11, and 12 to be used effectively.
Step 3: Fill Missing Nodes
Based on the remaining numbers:
One side includes the number 9 and needs additional numbers to reach the total of 21.
The possible pairs to complete the sides are:
Side 2: 9 + 12 = 21
Side 3: The remaining numbers to reach 21 are 11 and 10.
Completed Triangle Configuration
Top Node: 12
Left Node Below Top: 10
Right Node Below Top: 11
Magic Number: 21 for each side.

Ans: This calculation determines the correct arrangement of numbers from 4 to 9 in a magic triangle where each side totals to the same magic number.
Step 1: Sum of Numbers
The numbers available for use are 4, 5, 6, 7, 8, and 9, with their total sum calculated as follows:
Sum = 4 + 5 + 6 + 7 + 8 + 9 = 39
Step 2: Magic Number
As the triangle has three sides, the magic number, representing the total for each side, is:
Magic Number = Total Sum / Number of Sides = 39 / 3 = 13
Step 3: Configuring the Triangle
Given numbers in the triangle include 5, 7, 4, 8, and 6, with one number, 9, remaining to be placed:

• Top Node: Initially 5
• Left Node Below Top: Initially 4
• Right Node Below Top: Initially 7
• Bottom Left Node: Initially 8
• Bottom Right Node: Initially 6

The numbers must be arranged such that the sum of the numbers on each side equals 13:

• Side 1 (Top-Right): 5 + 7 (12) - Replace one number to include 9.
• Side 2 (Left-Bottom): 5 + 4 + 8 (17) - Exceeds 13, requires reconfiguration.
• Side 3 (Right-Bottom): 7 + 6 = 13 - Correct as is.

Final Configuration
After reconfiguring to fit the magic number, the triangle's numbers are set as follows to ensure each side totals 13:

• Top Node: 5
• Left Node Below Top: 4 (Moved to better fit the magic number)
• Right Node Below Top: 9 (Added to complete the configuration)
• Bottom Left Node: 8 (Remains unchanged)
• Bottom Right Node: 6 (Remains unchanged)
  
  

Question 3: Create a magic triangle using the numbers from 15 to 20.
Ans: Step 1: Calculate the Total Sum The numbers are 15, 16, 17, 18, 19, and 20. 
Sum = 15 + 16 + 17 + 18 + 19 + 20 = 105
Step 2: Determine the Magic Number
Magic Number = Total Sum / 3 = 105 / 3 = 35
Step 3: Arrange the Numbers
To create the magic triangle, we need to find sets of three numbers from 15 to 20 that sum to 35. Here’s one possible configuration:
Top Vertex: Use the middle value, say 18.
Other two vertices: Try to balance the two remaining sides to make each side total 35.
Given the chosen top vertex (18), the remaining numbers are 15, 16, 17, 19, 20. We can test combinations to achieve the magic sum on each side:
Pair with 18: 

• Attempt to use 16 and 17 along with 18: 18 + 16 + 17 = 51 (Doesn't work) 
• Another pair: 18 + 15 + 20 = 53  (Doesn't work) 
• 18 + 16 + 19 = 53 (Doesn't work) 
• We see that starting with 18 might be a challenge, so let’s choose a different top vertex.

Let’s rearrange: 

• New Top Vertex: Use a number that is easier to pair, say 17. 
• Calculate pairs: Now, 17 + 18 + ? 17+18+? must equal 35. The third number needs to be 35 − ( 17 + 18 ) = 0 35−(17+18)=0, impossible with our numbers. We need to switch to simpler pairings.

A more strategic approach: 

• Top Vertex: 17 
• Left Bottom Vertex: 15 (Pair it with 17 in a way that allows the last number to complete the triangle nicely) 
• Right Bottom Vertex: Find the pairing for 17 + 15 = 32, which leaves 3 to make 35. That’s not possible; let's recalibrate.

Since finding a direct solution can be complex, let's manually balance: 

• Top Vertex: 17 
• Left Bottom Vertex: 15 and 20 20 on one side: 17 + 15 = 32 (Need 3 more: use 18) 
• Right Bottom Vertex: 18 and 19 19 on the other side: 17 + 18 = 35 (Matches)

So, the pairing to ensure each side sums to 35: 

• Side 1: 17 + 15 + 18 = 50 (Reconfigure) 
• Side 2: 15 + 20 = 35 (Perfect) 
• Side 3: 17 + 18 = 35 (Perfect)

Final Configuration (needs further adjustment):

• Top: 17
• Bottom Left: 15
• Bottom Right: 18

Other possible configurations:

• Left: 19
• Right: 20

Question 4: Identify the magic numbers in the following magic triangles:

Ans: Triangle Configuration: 

• Top vertex: 7 
• Vertices below the top vertex: 9 (left) and 11 (right) 
• Bottom vertices: 8 (below 9), 10 (between 8 and 6), 6 (below 11)

Calculate Magic Number: We'll sum the numbers on each side and verify that they are equal to confirm the magic number: 

• Side 1: Combining 7, 9, and 8 = 7 + 9 + 8 = 24
• Side 2: Combining 7, 11, and 6 = 7 + 11 + 6 = 24
• Side 3: Combining 8, 10, and 6 = 8 + 10 + 6 = 24

The magic number for this triangle is 24, as all sides of the triangle sum to 24, maintaining the necessary balance for a magic triangle.

Ans: Triangle Configuration:

• Top vertex: 3
• Vertices below the top vertex: 7 (left) and 5 (right)
• Bottom vertices: 2 (below 7), 6 (between 2 and 4), 4 (below 5)

Calculate Magic Number: We'll sum the numbers on each side to verify if they are consistent: 

• Side 1: Combining 3, 7, and 2 = 3 + 7 + 2 = 12
• Side 2: Combining 3, 5, and 4 = 3 + 5 + 4 = 12
• Side 3: Combining 2, 6, and 4 = 2 + 6 + 4 = 12

The magic number for this triangle is 12, as all sides of the triangle sum to 12, confirming that it's a balanced magic triangle.

Question 5: Use numbers from 1 to 9 to create a magic square.
Ans: Steps to Create a 3x3 Magic Square:

• Start in the Middle of the Top Row: Place the number 1 in the middle cell of the top row.
• Move Up and Right: Move one step diagonally up and to the right to place the next number. If this move takes you outside the square, wrap around to the opposite side of the square. If a cell is already filled or you move outside the boundary, move down one cell instead.
• Follow the Rule: Continue placing the next higher number using the same rule: move up and right, wrapping around if necessary, and if the cell is filled or out of bounds, move down.

Here's how the numbers are placed:

• Place 1 in the middle of the top row.
• Move up and right for 2, but you go outside the square, so wrap around to the bottom left cell.
• For 3, move up and right from 2. You can wrap around to the last cell of the middle row.
• Continue this method to fill in 4, 5, 6, 7, 8, and 9.

Here's the filled magic square:

Explanation:

• Rows: (2 + 7 + 6), (9 + 5 + 1), (4 + 3 + 8) each sum to 15.
• Columns: (2 + 9 + 4), (7 + 5 + 3), (6 + 1 + 8) each sum to 15.
• Diagonals: (2 + 5 + 8), (6 + 5 + 4) each sum to 15.

Question 6: Identify the magic numbers for the magic squares shown below:

Ans: Top Row: 3 + 8 + 7 = 18 
Middle Row: 10 + 6 + 2 = 18
Bottom Row: 5 + 4 + 9 = 18
First Column: 3 + 10 + 5 = 18
Second Column: 8 + 6 + 4 = 18
Third Column: 7 + 2 + 9 = 18
Diagonal (Top-Left to Bottom-Right): 3 + 6 + 9 = 18
Diagonal (Top-Right to Bottom-Left): 7 + 6 + 5 = 18
Magic Number for the First Magic Square: 18

Ans: Top Row: 6 + 11 + 10 = 27 
Middle Row: 13 + 9 + 5 = 27
Bottom Row: 8 + 7 + 12 = 27 
First Column: 6 + 13 + 8 = 27
Second Column: 11 + 9 + 7 = 27
Third Column: 10 + 5 + 12 = 27 
Diagonal (Top-Left to Bottom-Right): 6 + 9 + 12 = 27
Diagonal (Top-Right to Bottom-Left): 10 + 9 + 8 = 27
Magic Number for the Second Magic Square: 27

Question 7: Create a magic square using the numbers from:

• 32 to 40

Ans: Start with the Middle Number: Find the middle of your number range, which is 36, and place it in the center of a 3x3 grid.
Place the Smallest Number: Start with the smallest number, 32, in the middle of the top row.
Follow a Specific Pattern: From 32, move up and to the right to place the next number. If this move takes you out of the grid, wrap around to the opposite side. If a cell is already filled, move directly below the last number you placed.
Continue this pattern until all numbers are placed.
Here’s a configuration that works:

• 7 to 15

Ans: Middle Number Placement: Place the middle number of your range, which is 11, in the center of the 3x3 grid.
Fill the Grid: Begin with the center and extend the sequence in a pattern that forms a cross, and continue filling in around the center systematically.
Pattern to Fill: Start with 11 in the center.
Fill the remaining cells such that every row, column, and diagonal sums to 33, the magic number for this set.
Correct Configuration: Here's a simple and accurate arrangement:

Adjust Each Number: Since the original configuration starts with numbers 1 to 9, add 6 to each number to convert them to the range 7 to 15:

Question 8: Complete the following patterns:

• (11 - 2) ÷ 9 = 1
  (111 - 3 ÷ 9 = 12
  (1111 - 4) ÷ 9 = 123
  _______________
  _______________
  _______________
  

Ans: (11 − 2) ÷ 9 = 1
(111 − 3) ÷ 9 = 12
(1111 − 4) ÷ 9 = 123
(11111 − 5) ÷ 9 = 1234
(111111 − 6) ÷ 9 = 12345
(1111111 − 7) ÷ 9 = 123456

• (2 x 2) - (1 x 1) = 3
  (3 x 3) - (2 x 2) = 5
  (4 x 4) - (3 x 3) = 7
  (5 x 5) - (4 x 4) = ______
  ______ - ______ = 11
  ( 7 x 7) - ( ______ ) = ______
  ( ______ ) - ( ______ ) = ______

Ans: (2 × 2) − (1 × 1) = 3
(3 × 3) − (2 × 2) = 5
(4 × 4) − (3 × 3) = 7
(5 × 5) − (4 × 4) = 9
(6 × 6) − (5 × 5) = 11
(7 × 7) − (6 × 6) = 13
(8 × 8) − (7 × 7) = 15

• 1 × 8 + 1 = 9
  12 × 8 + 2 = 98
  123 × 8 + 3 = 987
  1234 × 8 + 4 = ______
  ______ × 8 + ______ = 98765

Ans: 1 × 8 + 1 = 9
12 × 8 + 2 = 98
123 × 8 + 3 = 987
1234 × 8 + 4 = 9876
12345 × 8 + 5 = 98765
123456 × 8 + 6 = 987654

• 1 × 9 – 1 = 08
  21 × 9 - 1 = 188
  321 × 9 - 1 = 2888
  4321 × 9 – 1 = 38888
  54321 × 9 – 1 = ______
  654321 × 9 – 1 = ______
  ______ × 9 – 1 = 68888888

Ans: 1 × 9 − 1 = 08
21 × 9 − 1 = 188
321 × 9 − 1 = 2888
4321 × 9 − 1 = 38888
54321 × 9 − 1 = 488888
654321 × 9 − 1 = 5888888
7654321 × 9 − 1 = 68888888

Question 9: Complete the following patterns:
1 + 9 × 0 = 1
2 + 9 × 1 = 11
3 + 9 × 12 = 111
4 + 9 × 123 = 1111
5 + 9 × ______ = 11111
______ + 9 × ______ =______
7 + 9 × ______ =______
Ans: 1 + 9 × 0 = 1
2 + 9 × 1 = 11
3 + 9 × 12 = 111
4 + 9 × 123 = 1111
5 + 9 × 1234 = 11111
6 + 9 × 12345 = 111111
7 + 9 × 123456 = 1111111

Question 10: Complete the following magic triangles:
Ans: